Sheaf cohomology measures the obstruction of the global section functor from being exact. Since it's left exact, it is exact iff it preserves epis. In particular, $H^1$ measure the failure to be surjective.
The need to measure this failure makes sense - being able to lift sections locally is a much weaker condition than being to lift them globally, for instance due to topological considerations (e.g exponential sequence).
The fact $\Gamma$ preserves monos also makes sense since if one can locally differentiate between sections, then one may globally differentiate between them.
What bums me out is that unless I already know a functor is left/right exact, knowing it preserves epis/monos does not imply it's right/left exact. This is a bummer because I can't really think of any other intuitive ways to "suspect" whether some functor is half exact.
What is some geometric intuition behind left/right exactness? Sheafy examples are welcome, but I'd also like some general (hopefully geometric) intuition.
I don't know if geometry is the right place to look. The basic result is that a left adjoint is right exact (yeah, it's terrible that those don't match up), and dually a right adjoint is left exact. In fact left adjoints preserve all colimits (and right exact is equivalent to preserving all finite colimits), while right adjoints preserve all limits (and left exact is equivalent to preserving all finite limits).
The global sections functor is a right adjoint, which is why it's left exact. More generally, if $f : X \to Y$ is a continuous map, it induces an adjunction with left adjoint the pullback $f^{\ast} : \text{Sh}(Y) \to \text{Sh}(X)$ and right adjoint the pushforward $f_{\ast} : \text{Sh}(X) \to \text{Sh}(Y)$. Global sections is the special case of pushforward when $Y$ is a point.